Speculations of an amateur.

So I was watching the NOVA special, The Elegant Universe(and omg is it elegant), and they were talking about quantuum mechanics. Specifically what sparked the train of thought that I am about to detail was that in QM there is the certain possibility that a particle might actually travel through a wall, albeit rediculously small. This got me thinking about collision detection in games/real time simulations. Computers have the limiting factor of amount of time to process. Since this is the case, certain measures have to be taken that an object does not in fact travel through a wall. For example:

Take this particle a, and lets try to pass it through this wall |.

a-> |
| a

This is possible because by calculating the distance traveled from the last position of a, the velocity of a was great enough that at the time resolution available since the last known position of a, a would pass through the wall (to solve this you just draw a line between the two positions and use that to test the collision. I'm sure most the people here know that...).

This brings me back to QM. Why is there that possibility that the particle will pass through the wall? Is it because the resolution of time is not infinite (I am under the impression that it is)? Maybe there is a finite resolution to the distance a particle can travel at one instance due to its current velocity regardless of time.

I have absolutely no idea about quantuum mechanics. I just had a flash of supposed insight and thought I would ask the question. Please, let me know where I am wrong and why. I like being proven wrong. It elevates my understanding of things.

It has nothing to do with resolution, it is an intrinsic property of the particles. Their equation of motion are described by the Shrödinger equation, not newtons equations. The location of a particle is not even determined before one has measured it, so such questions are meaningless to ask in QM.

If you have never been introduced to quantum mechanics, then you cannot guess how it works. It is amazingly strange, and "no one in his right mind" would conceive a theory like that, if it weren't, initially, through a very convoluted path, that one was forced to it, and subsequently the amazing precision of its predictions.

In quantum mechanics, you give up thinking that a particle (a "system" more generally) is in a well-defined "state". Well, a particle ("system") has now a new set of "states" (which are "well-defined"), but which actually come down to having the particle ("the system") be in various "classical well-defined states" at the same time. Not that we are simply "ignoring" and hence have an "uncertainty" about its state (although things are sometimes presented that way, but you run into difficulties when you adhere to that view - at least without making *extra* assumptions, to keep the Bohmians happy) ; we say that the particle is in different classical states "at once".
If for instance, you consider "being in a well-defined geometrical position" a classically well-defined state of a particle, well, then quantum mechanics tells you that a quantum particle state comes down to having the particle be "at different places at the same time".

But worse, when you actually MEASURE the geometrical position of the particle, then you will find a classical result: you will find that it was in one particular position... with a certain probability. From experiment to experiment, you can prepare your particle in exactly the same "quantum state", and nevertheless, it will be measured to be in different locations each time, with a probability given by the "amount of presence" of that position in the "being in different states at once".

Now, of course, that's particularly weird, and for about 80 years now, there are debates about what does it actually mean (what is "really" happening).

Nevertheless, that's what the mathematical machinery of quantum mechanics does. But better, we have an equation (the famous Schroedinger equation) which lets us calculate how this "being at different positions at once" evolves through time, if we know how it was in the beginning, and if we know all of the interactions: that's a bit like Newton's equation, btw. If we know, in classical mechanics, where the particle was, and what are the interactions (forces), then we can calculate where it will be. Well, in quantum mechanics, that's similar, only, we don't calculate where the particle *is*, we calculate the "set of positions of where it will be at the same time and their importance".
This "set of positions it is at the same time and their weight" is also called the "particle wave function". So, the Schroedinger equation tells us how this wavefunction evolves through time. Because the Schroedinger equation looks a bit like a "wave equation" (of water waves), people called it in the beginning, the wave equation, and the solution, the "wavefunction". But in fact it is the catalog of "positions the particle is at the same time".

Well, it turns out that if you solve that equation for a (modeled) wall and an incoming particle, that the equation gives you a solution, after a while, where also positions *behind* the wall appear in the mix. Why ? Who knows. It comes out of the equation. Of course, there are mathematical reasons why such a type of equation gives such a type of solution, but that doesn't explain it physically.
Now, as we saw, if we *measure* the position of the particle, one of the "positions in the mix" comes about, and with a probability given by "the wave function squared". That means that there is a finite probability that the particle is actually detected behind the wall. Nobody really knows why, but it is what comes out of the wave equation.

Each time one had such a crazy result, and one was able to do an experiment verifying it, well, things turned out to be exactly as quantum theory predicted. That's why people study it, that's why they use it. It works. We know now mathematically *how* it works, but we still don't really know what is actually going on. It's crazy. But it works.

I would add (among the infinite number of other things one can add to this vast issue ) that the quantum "tunnelling" of particles through walls is not just speculations; two wires of aluminum (for example) put in contact together, conduct electric current for that reason. The metal infact has a protective layer of insulating oxide, which, classically, shoud prevent electric current to flow from one piece to the other.

Staff: Mentor

Well I haven't taken calculus(thats next semester), so I know that will be lost with the equations. What I am really trying to do is get the basic concepts of QM. I'm not trying to do research or anything. Yet. So a lamens explenation would probably be the best bet.

well I seldom recomend layman guides for QM, you wanted to understand the equations right?

Pictures and animations are good, but what is drawn is just the wave function, the formulas which are written down..

I tell you this, if you wait til after you have done some calculus, you will appreciate and understand quantum mechanics more. Learning QM without knowing math will probably make one even more confused when they encounter what QM really is.

Staff: Mentor

Second-year "modern physics" textbooks generally include an introduction to QM that is aimed at students who have just finished a first-year calculus-based introductory physics course. They don't go very deeply into the details, but they make it easier to start on a "real" undergraduate QM textbook such as Griffiths. They also give more information on the experimental basis and historical development of QM.